1
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1

Let the sixth term in the binomial expansion of $${\left( {\sqrt {{2^{{{\log }_2}\left( {10 - {3^x}} \right)}}} + \root 5 \of {{2^{(x - 2){{\log }_2}3}}} } \right)^m}$$ in the increasing powers of $$2^{(x-2) \log _{2} 3}$$, be 21 . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $$x$$ is __________.

2
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1

If the term without $$x$$ in the expansion of $$\left(x^{\frac{2}{3}}+\frac{\alpha}{x^{3}}\right)^{22}$$ is 7315 , then $$|\alpha|$$ is equal to ___________.

3
JEE Main 2023 (Online) 1st February Morning Shift
Numerical
+4
-1

The remainder, when $$19^{200}+23^{200}$$ is divided by 49 , is ___________.

4
JEE Main 2023 (Online) 31st January Evening Shift
Numerical
+4
-1
The coefficient of $x^{-6}$, in the

expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^{2}}\right)^{9}$, is